۱٫ Introduction

۲٫ Cyclic LCD Codes

۳٫ Cyclic LCD MDS Codes with Length

۴٫ LCD MDS Codes Derived from Cyclic Codes with Length

۵٫ Conclusion

۶٫ Acknowledgement

۷٫ References

Linear codes with complementary-duals (LCD) have many applications in cryptography, communication systems and data storage. A q-ary linear code C is called LCD if C∩C⊥ = {۰} holds. Using method of coset theory, we deduce a characterization of LCD cyclic code by its defining set. Then two families of q-ary MDS cyclic LCD codes with lengthes n| (q+1) and n| (q-1) are determined and many new classes of LCD MDS codes are gained.

Introduction

Refs [5-6] first present cyclic LCD codes and [4] constructed LCD MDS codes through Reed-Solomon (RS) codes and presented several methods of constructing LCD codes. Since then, researchers paid much attention to this topic and present many results, such as, LCD cyclic codes in [7-8], a condition that negacyclic codes are LCD codes in [9], LCD MDS codes through GRS codes in [11]. Inspired by [4,9-11], we study the methods of constructing LCD MDS codes from q-ary cyclic codes in this paper. This paper has five parts. In Section 2 we will provide some required basic knowledge on cyclic codes and LCD codes. We derive some classes of LCD codes with length n q ∣( ۱) and new LCD codes from these MDS codes in Section 3. In Section 4, we discuss LCD codes with n q ∣( ۱) . The last Section gives the conclusion.